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$a_n$ and $b_n$ are two bounden sequences Prove $$\liminf(a_n) + \liminf(b_n)\le \liminf(a_n +b_n)$$

Should I use $$\inf(a+b) = \inf(a) + \inf(b)$$ and i could not come up with how to proceed from this. Or should i use a different approach?

Martin Sleziak
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Bob
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  • @SuzuHirose it is the $\liminf$ http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior. – Crostul Nov 25 '14 at 12:21
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    liminf means limit infimum when I say $lim inf(a_n)$ I mean limit infimum of $a_n$ – Bob Nov 25 '14 at 12:21
  • @SuzuHirose, is the lim inf of a sequence . – Timbuc Nov 25 '14 at 12:23
  • @Bob, the original post wasn't clear at all. Also, those round parentheses didn't help. – Timbuc Nov 25 '14 at 12:23
  • Sorry about that I write the question from the phone. Edits will be appreciated – Bob Nov 25 '14 at 12:26
  • @Travis I don't agree that it is a duplicate, though it is closely related. One can mimick the proof there, or reduce the present case to the lim sup case by the judicious application of some minus signs. – Harald Hanche-Olsen Nov 25 '14 at 12:35
  • @HaraldHanche-Olsen Yes, one could well argue that. My view is that any student who knows enough analysis to the ask the question would immediately see the two questions as essentially identical. – Travis Willse Nov 25 '14 at 12:56
  • is the given inequality ,$$\liminf(a_n) + \liminf(b_n)\le \liminf(a_n +b_n)$$ valid only for bounded sequence,or also true for unbounded sequence? – math student Jul 09 '20 at 11:25

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Let us write $$ \liminf u_n = \sup_{N}\inf_{n>N}u_n $$

When you replace $u_n$ with $a_n + b_n$ you get successively: $$ m>N\implies a_m + b_m \ge \inf_{n>N} a_n + \inf_{n>N} b_n \\ \inf_{n>N} [a_n + b_n] \ge \inf_{n>N} a_n + \inf_{n>N} b_n \\ \inf_{n>N} [a_n + b_n] \ge \sup_N\inf_{n>N} a_n + \inf_{n>N} b_n\\ \sup_N\inf_{n>N} [a_n + b_n] \ge \sup_N\inf_{n>N} a_n + \inf_{n>N} b_n $$

mookid
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